Displaying similar documents to “On a property of neighborhood hypergraphs”

Set vertex colorings and joins of graphs

Futaba Okamoto, Craig W. Rasmussen, Ping Zhang (2009)

Czechoslovak Mathematical Journal

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For a nontrivial connected graph G , let c V ( G ) → ℕ be a vertex coloring of G where adjacent vertices may be colored the same. For a vertex v of G , the neighborhood color set NC ( v ) is the set of colors of the neighbors of v . The coloring c is called a set coloring if NC ( u ) ≠ NC ( v ) for every pair u , v of adjacent vertices of G . The minimum number of colors required of such a coloring is called the set chromatic number χ s ( G ) . A study is made of the set chromatic number of the join G + H of two graphs G and H . Sharp lower...

Maximum Edge-Colorings Of Graphs

Stanislav Jendrol’, Michaela Vrbjarová (2016)

Discussiones Mathematicae Graph Theory

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An r-maximum k-edge-coloring of G is a k-edge-coloring of G having a property that for every vertex v of degree dG(v) = d, d ≥ r, the maximum color, that is present at vertex v, occurs at v exactly r times. The r-maximum index [...] χr′(G) χ r ' ( G ) is defined to be the minimum number k of colors needed for an r-maximum k-edge-coloring of graph G. In this paper we show that [...] χr′(G)≤3 χ r ' ( G ) ≤ 3 for any nontrivial connected graph G and r = 1 or 2. The bound 3 is tight. All graphs G with [...] χ1′(G)=i...

On k-intersection edge colourings

Rahul Muthu, N. Narayanan, C.R. Subramanian (2009)

Discussiones Mathematicae Graph Theory

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We propose the following problem. For some k ≥ 1, a graph G is to be properly edge coloured such that any two adjacent vertices share at most k colours. We call this the k-intersection edge colouring. The minimum number of colours sufficient to guarantee such a colouring is the k-intersection chromatic index and is denoted χ’ₖ(G). Let fₖ be defined by f ₖ ( Δ ) = m a x G : Δ ( G ) = Δ χ ' ₖ ( G ) . We show that fₖ(Δ) = Θ(Δ²/k). We also discuss some open problems.

Hajós' theorem for list colorings of hypergraphs

Claude Benzaken, Sylvain Gravier, Riste Skrekovski (2003)

Discussiones Mathematicae Graph Theory

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A well-known theorem of Hajós claims that every graph with chromathic number greater than k can be constructed from disjoint copies of the complete graph K k + 1 by repeated application of three simple operations. This classical result has been extended in 1978 to colorings of hypergraphs by C. Benzaken and in 1996 to list-colorings of graphs by S. Gravier. In this note, we capture both variations to extend Hajós’ theorem to list-colorings of hypergraphs.

New edge neighborhood graphs

Ali A. Ali, Salar Y. Alsardary (1997)

Czechoslovak Mathematical Journal

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Let G be an undirected simple connected graph, and e = u v be an edge of G . Let N G ( e ) be the subgraph of G induced by the set of all vertices of G which are not incident to e but are adjacent to u or v . Let 𝒩 e be the class of all graphs H such that, for some graph G , N G ( e ) ≅ H for every edge e of G . Zelinka [3] studied edge neighborhood graphs and obtained some special graphs in 𝒩 e . Balasubramanian and Alsardary [1] obtained some other graphs in 𝒩 e . In this paper we given some new graphs in 𝒩 e .

Kaleidoscopic Colorings of Graphs

Gary Chartrand, Sean English, Ping Zhang (2017)

Discussiones Mathematicae Graph Theory

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For an r-regular graph G, let c : E(G) → [k] = 1, 2, . . . , k, k ≥ 3, be an edge coloring of G, where every vertex of G is incident with at least one edge of each color. For a vertex v of G, the multiset-color cm(v) of v is defined as the ordered k-tuple (a1, a2, . . . , ak) or a1a2 … ak, where ai (1 ≤ i ≤ k) is the number of edges in G colored i that are incident with v. The edge coloring c is called k-kaleidoscopic if cm(u) ≠ cm(v) for every two distinct vertices u and v of G. A regular...

Contractible edges in some k -connected graphs

Yingqiu Yang, Liang Sun (2012)

Czechoslovak Mathematical Journal

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An edge e of a k -connected graph G is said to be k -contractible (or simply contractible) if the graph obtained from G by contracting e (i.e., deleting e and identifying its ends, finally, replacing each of the resulting pairs of double edges by a single edge) is still k -connected. In 2002, Kawarabayashi proved that for any odd integer k ≥ 5 , if G is a k -connected graph and G contains no subgraph D = K 1 + ( K 2 ∪ K 1 , 2 ) , then G has a k -contractible edge. In this paper, by generalizing this result, we prove that...